PRICING DOUBLE BARRIER OPTIONS ON HOMOGENEOUS DIFFUSIONS: A NEUMANN SERIES OF BESSEL FUNCTIONS REPRESENTATION

This paper develops a novel analytically tractable Neumann series of Bessel functions representation for pricing (and hedging) European-style double barrier knock-out options, which can be applied to the whole class of one-dimensional time-homogeneous diffusions, even for the cases where the corresponding transition density is not known. The proposed numerical method is shown to be efficient and simple to implement. To illustrate the flexibility and computational power of the algorithm, we develop an extended jump to default model that is able to capture several empirical regularities commonly observed in the literature.

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CLOSED FORM FORMULAS FOR EXOTIC OPTIONS AND THEIR LIFETIME DISTRIBUTION

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024999000030 ◽

1999 ◽

Vol 02 (01) ◽

pp. 17-42 ◽

Cited By ~ 22

Author(s):

RAPHAËL DOUADY

Keyword(s):

Interest Rates ◽

Closed Form ◽

Term Structure ◽

Exit Time ◽

Exotic Options ◽

Barrier Options ◽

Double Barrier ◽

Knock Out ◽

The Mean ◽

Mirror Principle

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge like e-α n2.

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Pricing discrete double barrier options with a numerical method

Journal of Asset Management ◽

10.1057/jam.2015.6 ◽

2015 ◽

Vol 16 (4) ◽

pp. 243-271

Author(s):

Pierre Rostan ◽

Alexandra Rostan ◽

François-Éric Racicot

Keyword(s):

Numerical Method ◽

Barrier Options ◽

Double Barrier

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A Numerical Method for Pricing Discrete Double Barrier Option by Lagrange Interpolation on Jacobi Nodes

10.22541/au.162365543.38524042/v1 ◽

2021 ◽

Author(s):

Amirhossein Sobhani ◽

mariyan milev

Keyword(s):

Numerical Method ◽

Lagrange Interpolation ◽

Integral Formula ◽

Operational Matrix ◽

Barrier Option ◽

Double Barrier ◽

Knock Out ◽

Black Scholes Model ◽

Important Benefit ◽

Black Scholes

In this paper, a rapid and high accurate numerical method for pricing discrete single and double barrier knock-out call options is presented. With regard to the well-known Black-Scholes model, the price of an option in each monitoring date could be calculated by computing a recursive integral formula that is based on the heat equation solution. We have approximated these recursive solutions with the aid of Lagrange interpolation on Jacobi polynomial nodes. After that, an operational matrix, that makes our computation significantly fast, has been derived. In some theorems, the convergence of the presented method has been shown and the rate of convergence has been derived. The most important benefit of this method is that its complexity is very low and does not depend on the number of monitoring dates. The numerical results confirm the accuracy and efficiency of the presented numerical algorithm.

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